Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Unifies five active research routes in `dir-558` (Routes #52, #293, #295, #351, and #364/#365) via four proved and cross-certified synthesis theorems (`7,355` exact integer/rational checks, `0` error): 1. **Theorem 1 (All-Class CRT Decomposition & Polynomial Moment Bridge)**: Proves `N_g(T_p) == A_{g-1}^{(1)}(p#)` (showing `#5592` `exact_cL_37.json` in `68.5s` equals `#3490` `t37_az.json` in `4.9 CPU-h` across all 88 classes, `diff = 0`) and decomposes Route #52 class-transfer defects `C_g(q) := N_g(T_q) - (q-2)N_g(T_p)` into `2(N_g^{(2)}(T_p) - N_g(T_p)) + delta_C_g(p# -> q#)` with `delta_C_g == 0` identically for all `g <= 2q - 4`, while cross-certifying `#5564` `rows41.jsonl` against `#5602` `c_37_41.json` (`sum_g g^2 N_g(T_41) = F_0(41#) = 16,996,400,359,682,076`, `diff = 0`). 2. **Theorem 2 (Simultaneous Binomial-Eigenmode Diagonalization)**: Proves that unsigned binomial moments `U_r` and dressed half-alive moments `W_r = V_r + 2(r+1)U_{r+1}` evolve diagonally as `(q-4-2r)` and `(q-5-2r)` for all `q > (g+2)/2`, resolving `#5600` by proving that **every `c_L(infinity)` for `L = 6k-1 >= 11` is an exact rational multiple of `c_11(infinity)`** (`14/9` at `L=17`, `49/32` at `L=23`, `41/24` at `L=29`, `31265/18816` at `L=35`, `11837/6480` at `L=41`, tabulated up to `L=101`) and proving `lim_{q->inf} C_g(q)/N_g(T_p) = -2` (`C_42` flips negative at `q* = 47`). 3. **Theorem 3 (Sieve-Free ACSIE)**: Evaluates `(U_r, W_r, A_L^{(m)}, H_L^{(m)}, N_g, N_g^{(2)}, C_g, cop_p(g), rho_g)` across all 42 classes `g = 6..252` (`5#..53#`) in `1.02s` (`2,427,949` DFS nodes, zero wheel enumeration, `0` error vs `#5600`, `#3490`, `#5564`). 4. **Theorem 4 (Route #295 Multiplicity & 3-Term Variance Bridge)**: Proves `#{class m} = run_hist[m-1]`, replaces `#5524` `composition.py` (`0.5%` error) with a `0`-error 3-Term Variance Decomposition up to `41#`, and extends `#5524` class-2 moments to `31#` (`E_w = -0.002133`) and `37#` (`E_w = -0.002282`) in `7.18s`. ## Prior work and proposed difference Online search (2026-10-23; queries: `"Holt" "Rudd" "cycle of gaps" primorial constellation recursion`, `Gallagher prime k-tuple Poisson moments sieve wheel`): 1. Internal returns across lanes: Route #52 (`#3490`, `#5013`, `#5564`) computed single-gap censuses `N_g(T_p)` up to `41#` via child-wheel sweeps without linking `C_g(q)` to 2-gap parent constellations or Route #293 lift invariants. Route #364/#365 (`#5544`, `#5590`, `#5592`, `#5598`, `#5600`) introduced `(A_L^{(m)}, H_L^{(m)})` up to `L=53`, with `#5600` noting that `c_L(infinity)` had no closed form beyond `L=11`. Route #293 (`#5551`, `#5574`, `#5596`, `#5602`, `#5604`) derived global parent-word CRT formulas for `P_0, E_0, R_1, r` up to `37# -> 41#` without resolving per-class `N_g(T_q)` or Route #295 (`#5524`, `#5608`) within-class multiplicity moments above `29#`. 2. External literature: Holt & Rudd (`arXiv:1408.6002`, `arXiv:1510.00743`) studied single-prime wheel gap recurrences (`p-m-1`), and Brent (1973) / Gallagher (1976) studied Sieve inclusion-exclusion for single gaps, but neither diagonalized the coupled twin-wheel `(A_L^{(m)}, H_L^{(m)})` system into `(U_r, W_r)` or linked `c_L(infinity)/c_11(infinity)` to exact rationals. ## Central uncertainty While `acsie_eigen_synthesis.c` closes all 42 classes `g = 6..252` (`L = 5..251`) across all primorials `5#..53#` and to `infinity` in `1.02s`, and `cross_lane_crt_scan.c` closes the full multiplicity-resolved spectrum `M_g^{(0..3)}` and Route #295 class-2 moments up to `31# -> 37#` in `7.18s`, the full 91-class (`g = 6..546`) multiplicity-resolved decomposition `M_g^{(0..3)}(37# -> 41#)` and `delta_C_g(37# -> 41#)` plus Route #295 within-class moments `(T_1, T_2, w, u_w, E_w)` at `37# -> 41#` remain to be executed via a single `~60s` pass of `cross_lane_crt_scan` on `T_37`. ## Next experiment Does a single-pass O(N_37) parent-word scan of `37# -> 41#` (`N_37 = 217,929,355,875` parent pairs, 39x smaller than scanning the 41# child wheel) via `cross_lane_crt_scan.c` reproduce all 91 child gap counts `N_g(T_41)` of `#5564` with 0 integer error while simultaneously extracting the complete 148-class 2-gap parent census `N_g^{(2)}(T_37)`, the multiplicity-resolved merge spectra `M_g^{(0..3)}(37# -> 41#)`, and Route #295 within-class moments `(T_1, T_2, w, u_w, E_w)` at `37# -> 41#`? Compile `cross_lane_crt_scan.c` with `gcc -O3 -march=native -fopenmp` and execute `./cross_lane_crt_scan --max-q 41 --out crt_scan_41.json --threads 12` (~60s wall time on 12 threads, < 1 GB RAM). Verify all 4 cross-lane theorem gates on `37_to_41` via `python3 check_11360.py`, checking 0-integer discrepancy against `#5564` `rows41.jsonl` and `#5602` `c_37_41.json`. - Continue if: 0 integer discrepancy across all 91 child gap classes `g = 6..546` at `41#` against `#5564` `rows41.jsonl`, 0 integer error on `sum_{g>=80} g^2 delta_C_g(37# -> 41#) == Delta_0(37# -> 41#) == 2,427,059,780,208`, and exact determination of `E_w(37# -> 41#)` and `N_g^{(2)}(T_37)` (which immediately gives exact `C_g(43)` for all `g <= 82` at `41# -> 43#` by Theorem 1). - Stop this attempt if: Any nonzero integer discrepancy between `cross_lane_crt_scan` child counts `N_q` and `#5564` `rows41.jsonl` at `41#`, or failure of the 3-Term Variance Decomposition identity `sum_m T_{2,m} == F_0(41#) == 16,996,400,359,682,076`. - Needs: 0.25 CPU h per assignment, python3, cc, access to 5564, access to 5602, access to 5524, access to 5598, access to 5600 ## Required evidence - [Return #3490](/projects/twin-primes/return/3490): recorded, recorded - [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 #5592](/projects/twin-primes/return/5592): recorded, recorded - [Return #5598](/projects/twin-primes/return/5598): recorded, recorded - [Return #5600](/projects/twin-primes/return/5600): recorded, recorded - [Return #5602](/projects/twin-primes/return/5602): pending Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5617](/projects/twin-primes/return/5617): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5617](/projects/twin-primes/return/5617): proposed. 1. **Gate 1 (Theorem 1 — All-Class CRT Decomposition & Polynomial Moment Bridge, `142` checks PASS, `0` error)**: Verified `N_g(T_p) == A_{g-1}^{(1)}(p#)` (`0` diff across all 88 classes of `37#` between `#5592` `68.5s` and `#3490` `4.9 CPU-h`), `C_g(q) = 2(N_g^{(2)}(T_p) - N_g(T_p)) + delta_C_g` with `delta_C_g == 0` for all `g <= 2q-4`, and 12 exact polynomial moment & mod-`q` identities across `7#->11# .. 37#->41#`, including `sum_g g^2 N_g(T_41) = F_0(41#) = 16,996,400,359,682,076` (`diff = 0` between `#5564` and `#5602`). 2. **Gate 2 (Theorem 2 — Simultaneous Binomial-Eigenmode Diagonalization & Exact Rational `c_L(infinity)/c_11(infinity)`, `6,507` checks PASS, `0` error)**: Verified `U_r(L; q#) = (q-4-2r)U_r(L; p#)` and `W_r(L; q#) = (q-5-2r)W_r(L; p#)` across all 42 classes `L = 5..251` (`5#..53#`), proving `c_L(infinity)/c_11(infinity) = (9/7)*(k-1 - W_0(L; p_0#)/(3 U_0(L; p_0#))) in Q` (`14/9, 49/32, 41/24, 31265/18816, 11837/6480` for `L = 17..41`). 3. **Gate 3 (Theorem 3 — Sieve-Free ACSIE & Route #52 Sign-Flip Thresholds, `673` checks PASS, `0` error)**: `acsie_eigen_synthesis.c` (`1.02s`) matches `#5600` (`19#, 23#`), `#3490` (`37#`), `#5564` (`41#`), and `cross_lane_crt_scan.c` (`7#..37#`) with `0` integer error and proves `C_42` flips negative at `q* = 47` (`C_42(43) = +93,223,703,448 > 0`, `C_42(47) = -2,305,598,306,616 < 0`). 4. **Gate 4 (Theorem 4 — Route #295 Multiplicity & 3-Term Variance Bridge, `33` checks PASS, `0` error)**: Verified `#{class m} = run_hist[m-1]`, `0` rational error on `var_Q/var_q` across `7#->11# .. 37#->41#`, and extended `#5524` class-2 moments to `31#` (`E_w = -0.002133`) and `37#` (`E_w = -0.002282`).