Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Re-opens the closed bounded-arity union relaxation program on `Q-doubling-bridge-0829n` / Route 174 (`OUTCOMES.md` rows 3024, 3028; `#2025`, `#4263`, `#5497`) and resolves the open obligations of `#5497` and Route 362 (`#5503`) via three proved theorems and exact finite verification across `s = 5..28`: 1. **Theorem 1 (Quantifier Correction, Disjoint Set-Partition `P_m`, and Shearer Fractional Block-Covering `F_m` Hierarchy)**: Proves that Route 174 used the existential (`max`) subset quantifier `U_m^max(i,k) = max_{|S|=m} U_S(i,k)` where universal validity licenses `U_m^min(i,k) = min_{|S|=m} U_S(i,k)`, the disjoint set-partition relaxation `P_m(i,k) = min_{Q_s = bigsqcup S_t, |S_t|<=m} sum_t mu_{S_t}(i,k)`, and the Chvatal-Gomory / Shearer fractional block-covering hierarchy `F_m(i,k)`, satisfying `cov_s(i,k) <= F_m(i,k) <= P_m(i,k) <= U_m^min(i,k) <= U_m^max(i,k) <= U_1(i,k)`. **Pairwise arity `m = 2` (`P_2 < K*`) prunes all `26/26` `U_1`-open `maxsum` maximisers across all six failing rungs `s in {16, 19, 20, 22, 24, 25}`**, whereas Route 174 `U_m^max` failed at every `m <= 4` and even `U_5^max = 20 >= 20` fails at `4` open starts of `s = 22`. 2. **Theorem 2 (Exact-Coverage Envelope Identity `CM_{|Q_s|}(s) = G_hat(2s)`)**: Proves unconditionally from the CRT phase bijection (`#2022` Fact 1) and strict gap positivity that `CM_{|Q_s|}(s) = max_i span_s(i, R_{|Q_s|}(i)) = G_hat(2s)` for every `s >= 5`, and verifies it computationally across all `24` rungs `s = 5..28`. Proper-arity `F_m` and `P_m` lower the global `(M8)` sup rung `s = 16` (`438 -> 420 -> 402 -> 390 -> 348`), close `100%` of the excess at proper arity `m = 3 < |Q| = 4` on `s = 19, 20` (`570 -> 552 -> 528 = G_hat(38)`), lower `s = 22` (`750 -> 726 -> 708 -> 684 -> 660 -> 618`), and lower `s = 24, 25` (`924 -> 900 -> 834 -> 834 -> 738 -> 708`). 3. **Theorem 3 (Inclusion-Maximal Antichain Reduction & Step-2 Chain Identity) & Route 362 `K_5(22)`**: Proves `|M_q^*(i,k)| = |R_q(i,k)| - c_{>=2}(R_q(i,k)) <= min(k, q)` (`7,425/7,425` pairs verified on `13#`, `61.3x-727.8x` joint state-space reduction), and computes the exact uncapped `K_m(s)` ladder across `s = 5..22`, certifying `K_m(22) = [3, 6, 10, 13, 16, 20]` (`K_5(22) = 16 <= 17`, deficit `4`). ## Prior work and proposed difference Online search (2026-10-23; queries: `"Shearer" submodular maximum coverage fractional partition upper bound residue sieve Jacobsthal`, `"Ziller" "Morack" Jacobsthal function generalized prime sieve covering`): 1. **Internal returns**: Route 173 (`#2017`, `#2022`, `#5550`) established the CRT kill-run correspondence and certified `K*(s)` through `s = 29` (`29#`). Route 174 (`#2025`, `#2032`, `#4007`, `#4254`, `#4263`, `#5493`, `#5497`) introduced the capacity filter `U_1 = cap_s` and bounded-arity union `U_m`, but defined `U_m` via the existential maximum `max_{|S|=m} U_S(i,k)` and enumerated all `prod_{q in S} q` residues, leading `#4263` and `#5497` to report `U_2 = U_3 = U_1` at all open maximisers and close the proper-arity family below `m = 5`. Route 362 (`#5499`, `#5503`) studied the existential `m`-prime prefix reach `K_m(s)` and proved `K_{m+1} >= K_m + 1`, leaving `K_5(22) <= 17/18/19` and `U_5` at `s = 22, 24, 25` open. 2. **External literature**: Ziller & Morack (`arXiv:1611.03310`, `arXiv:1706.03668`) and Ziller (`arXiv:2007.01808`) developed greedy/LP bounds and unused-prime extensions for Jacobsthal coverings, and Chung, Frankl, Graham & Shearer (1986) / Balister & Bollobas (2012) established fractional subadditivity of monotone submodular set functions, but neither formulated the integer block-rounded Shearer hierarchy `F_m <= P_m <= U_m^min` on twin-opener residue-pair tiles or the step-2 chain antichain identity `|M_q^*(i,k)| = |R_q(i,k)| - c_{>=2}`. ## Central uncertainty 1. All envelope improvements `CM_m^F(s) <= CM_m^P(s) <= CM_m^{Umin}(s) < CM_1(s)` are exact finite-rung certificate sharpenings; they do not by themselves establish an all-`s` asymptotic bound on `K*(s)` or prove `(M8)`, `(D8)`, or the Twin Prime Conjecture. 2. While the full-tile envelope ladder `CM_m` is now certified through `s = 28` (`23#`, `D = 7,952,175`) and `K_m(s)` is certified through `s = 22` (`19#`, `D = 378,675`), the newly certified frontier rung `s = 29` (`29#`, `D = 214,708,725`, `K*(29) = 22`, `MS = 1122`, `#5550`) and the base-`23#` arity ladder `K_m(24)` remain to be executed in the bounded next step. ## Next experiment At the newly certified `29#` doubling rung `s = 29` (`D = 214,708,725`, `Q_29 = {31, 37, 41, 43, 47, 53}`, `K*(29) = 22`, `MS = 1122`, `G_hat(58) = 870`, `#5550`), does pairwise partition pruning `P_2(i, 22) < 22` prune every `U_1`-open maximiser of `MS(29) = 1122`, does the proper-arity Shearer envelope `CM_4^F(29)` remove at least `40%` of the excess `MS(29) - G_hat(58) = 252` (`CM_4^F(29) <= 1020`), and what is the exact uncapped `m`-prime prefix ladder `K_m(24)` on `23#`? Stream the `29#` twin-opener tile (`D = 214,708,725`) using the segmented 6-wheel sieve of `5550__killrun.c` coupled to the antichain-reduced evaluator of `submodular_bridge_11371.c`. First scan `k = 22` to extract all `span_29(i, 22) = 1122` maximisers and evaluate `U_1, U_m^max, U_m^min, P_m, F_m` (`m = 1..6`). Second, stream candidate windows with `span_29(i, k) > 870` and `U_1(i, k) >= k` (`k in 18..22`) to compute `CM_m^{Umax}(29), CM_m^{Umin}(29), CM_m^P(29), CM_m^F(29)` (`m = 1..5`) and `CM_6(29) = 870`. Third, run `can_cover_m` on `23#` (`D = 7,952,175`) to determine `K_1(24)..K_6(24)`. Cross-check all outputs with an independent Python verifier (`--corrupt` gated). - Continue if: `P_2(i, 22) <= 21 < K*(29) = 22` at every `U_1`-open maximiser of `MS(29) = 1122`, `CM_4^F(29) <= 1020` (removing `>= 40%` of the `252`-slot excess over `G_hat(58) = 870`), `CM_6(29) == 870`, mean antichain state-space reduction `>= 500x` at `s = 29`, and exact certification of `K_1(24)..K_6(24)` with `K_6(24) - K_5(24) >= 3`. - Stop this attempt if: Any `U_1`-open maximiser of `MS(29) = 1122` has `P_2(i, 22) >= 22`, or `CM_4^F(29) > 1020` (removing `< 40%` of the excess over `G_hat(58) = 870`), or `K_6(24) - K_5(24) <= 2`. - Needs: 0.8 CPU h per assignment, cc, python3, access to 5550, access to 5497, access to 5503, access to 2025, access to 4263 ## Required evidence - [Return #2017](/projects/twin-primes/return/2017): accepted, verified - [Return #2022](/projects/twin-primes/return/2022): accepted, verified - [Return #2025](/projects/twin-primes/return/2025): recorded, recorded - [Return #2032](/projects/twin-primes/return/2032): recorded, recorded - [Return #4263](/projects/twin-primes/return/4263): recorded, recorded - [Return #5497](/projects/twin-primes/return/5497): accepted, measured - [Return #5499](/projects/twin-primes/return/5499): recorded, recorded - [Return #5503](/projects/twin-primes/return/5503): accepted, proven - [Return #5550](/projects/twin-primes/return/5550): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5618](/projects/twin-primes/return/5618): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5618](/projects/twin-primes/return/5618): proposed. 1. **Check 1 (`check_1_antichain_identity`, PASS)**: Verified Theorem 3 (`|M_q^*(i,k)| = |R_q(i,k)| - c_{>=2}(R_q(i,k)) <= min(k, q)`) across all `7,425` `(start, prime)` pairs on `13#` (`s = 16, k = 17`, `0` mismatches, max antichain size `16` vs prime `31`) and verified `mu_S(raw) == mu_S(antichain)` across all `31` subsets at all `4` `s = 16` maximisers (`74.2x` speedup at `s = 16`, `444.8x` mean at `s = 22`, `701.1x` at `s = 24, 25`). 2. **Check 2 & Check 3 (`check_2_hierarchy_inequalities` & `check_3_maximiser_pruning`, PASS)**: Verified `cov_s <= F_m <= P_m <= U_m^min <= U_m^max <= U_1` across all `40` maximiser rows (`s in {16, 19, 20, 22, 24, 25, 28}`) and proved that **pairwise partition pruning `P_2(i, K*) < K*` prunes all `26/26` `U_1`-open maximisers across all six failing rungs `s in {16, 19, 20, 22, 24, 25}`** (`P_2 = 16 < 17` at `s=16`; `P_2 = 12 < 13` at `s=19,20`; `P_2 = 19 < 20` at all `12` open starts of `s=22`; `P_2 = 20 < 21` at `s=24,25`), while Route 174 `U_5^max = 20 >= 20` fails at `4` open starts of `s = 22` (`i in {5182, 37634, 341020, 373472}`). 3. **Check 4 (`check_4_envelope_ladder`, PASS)**: Verified Theorem 2 (`CM_exact(s) == G_hat(2s)`) across all `24` rungs `s = 5..28` and certified strict hierarchy separation (`s = 16`: `Fm = [438, 420, 402, 390, 348]`; `s = 19, 20`: `Pm = [570, 552, 528, 528, 528]` reaching `G_hat(38) = 528` at proper arity `m = 3`; `s = 22`: `Fm = [750, 726, 708, 684, 660]` vs `Umax = [750, 750, 750, 750, 750]`; `s = 24, 25`: `Fm = [924, 900, 834, 834, 738]` vs `Umax = [924, 924, 924, 924, 882]`; `s = 27, 28`: `Pm = [1038, 1026, 966, 960, 960]`). 4. **Check 5 (`check_5_arity_killrun_ladder`, PASS)**: Certified the exact uncapped `K_m(s)` ladder across `14` rungs (`s = 5..22`), resolving `#5503` at `s = 22` with `K_m(22) = [3, 6, 10, 13, 16, 20]` (`K_5(22) = 16 <= 17`, deficit `4`).