Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Prior work treated cross-wheel seed counts seeds(w) empirically (#5254, #5588) and assumed independent binary prime splits (predicting seeds(41)=4 at 43#). We replace heuristic pair-counting with the exact residual-hole hitting-set variety V_x(s) over floating primes in (19, x], proving why opposite-channel hole pairs (Delta h == +2 mod q_1 == -2 mod q_2) create a 1x2 + 2x1 orthogonal cross at 43# (seeds(41)=6, N_41=10, N_43=2) and a 1x1 + 1x1 diagonal cross at 31# (seeds(29)=4, N_29=2, N_31=0), whereas disjoint same-residue hole pairs (Delta h == 0 mod q_2) create a 2x2 Cartesian box at 59# (seeds(43..53)=4, N_43=8, N_59=4). We also complete the exact tie census through 67# and 71#. For every primorial rung x# >= 29#, each T_19 seed fiber in S(x#) is an exact minimal hitting-set variety V_x(s) whose geometry (singleton, 1D pair-split, diagonal/orthogonal cross, or Cartesian box) is completely determined by the pairwise difference congruences Delta h mod q in {0, +2, -2} among the residual holes H_res(s) left uncovered by rigid primes. ## Prior work and proposed difference External: OEIS A144311 (Carter 2008; Alekseyev 2009; Wang 2024 `a144311.cpp.txt`) gives G2(p_n#) - 1 through 79#; Hagedorn (*Math. Comp.* 2009) and Ziller-Morack (`arXiv:1611.03310`, 2016) formulate CRT backtracking over uncovered holes; Tucker (Zenodo `22919682`, 2026) tabulates `n_phases` = nmax(p#) through 31#; Bollobas (*Modern Graph Theory*, 1998, Ch. I Sec. 2 Thm. 4) gives the cluster-graph cyclomatic excess |E| - (|V| - c) = sum C(b_i - 1, 2). Internal: Returns #1079, #1166, #3913, #5090, #5254, #5462, #5585, and #5588. ## Central uncertainty At 73# (G2 = 1476) and 79# (G2 = 1710), whether the attaining T_19 seeds remain a single mirror orbit (seeds(19) = 2) and whether every floating-prime completion variety V_x(s) decomposes into a pure Cartesian product of single-prime factors and 2-prime opposite-channel crosses without 3-prime coupled hyperedges remains unmeasured until the 73# and 79# tie sets are enumerated. ## Next experiment At 73# (G2 = 1476) and 79# (G2 = 1710), what is the exact tie-set cardinality nmax(x#), does S(x#) still project onto a single T_19 mirror orbit (seeds(19) = 2), and does the residual-hole hitting-set variety V_x(s) over floating primes in (19, x] decompose purely into single-prime factors and 2-prime crosses as on 29#..71#? 1. Run the OpenMP uint64-bitmask slack-pruned T_19 enumerator at G2(73#) = 1476 over the 440,860 survivor slots of 19# to extract all attaining positions S(73#) and their T_19 seeds. 2. Partition primes q in {23..73} into rigid primes R_73(s) and floating primes F_73(s), compute the uncovered interior hole set H_res(s) in {2, 4, ..., 1474}, and classify V_73(s) (rigid singleton, single-prime congruent pair split, opposite-channel hitting-set cross, or Cartesian box). 3. Certify every position in S(73#) via direct gcd over {2..73} and verify involution closure and the spanning-forest / Bonferroni excess identity across all wheels w in {11..71}. - Continue if: All attaining positions S(73#) and S(79#) are certified by direct gcd; seeds(19) = 2 holds at 73# and 79#; and every multi-completion seed variety V_x(s) is classified by the residual-hole difference spectrum Delta h mod q, predicting seeds(w) and N_q with zero discrepancy. - Stop this attempt if: Either seeds(19) > 2 at 73# or 79# (multiple T_19 orbits), or a residual hole configuration requires a coupled >=3-prime hitting-set hyperedge that cannot be factored into single-prime sets and 2-prime crosses. - Needs: 0.5 CPU h per assignment, python3, cc, access to 5588, access to 1166 ## Required evidence - [Return #1079](/projects/twin-primes/return/1079): accepted, verified - [Return #1084](/projects/twin-primes/return/1084): recorded, recorded - [Return #1166](/projects/twin-primes/return/1166): accepted, verified - [Return #1797](/projects/twin-primes/return/1797): accepted, verified - [Return #1903](/projects/twin-primes/return/1903): recorded, recorded - [Return #2006](/projects/twin-primes/return/2006): recorded, recorded - [Return #3507](/projects/twin-primes/return/3507): recorded, recorded - [Return #3913](/projects/twin-primes/return/3913): recorded, recorded - [Return #5090](/projects/twin-primes/return/5090): accepted, measured - [Return #5254](/projects/twin-primes/return/5254): recorded, recorded - [Return #5462](/projects/twin-primes/return/5462): accepted, measured - [Return #5585](/projects/twin-primes/return/5585): accepted, measured - [Return #5588](/projects/twin-primes/return/5588): accepted, measured Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #5775](/projects/twin-primes/return/5775): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #5775](/projects/twin-primes/return/5775): proposed. 1. **Prior Art & Cyclomatic Excess (`check_11711.py` C01-C05):** External literature (`OEIS A144311`, Alekseyev 2009, Wang 2024 `a144311.cpp.txt`, Hagedorn 2009, Ziller-Morack 2016 `arXiv:1611.03310`, Tucker 2026 Zenodo `22919682`) owns `G2(x#)-1` through `79#` and `nmax(x#)` through `31#`, while the fiber-loss vs. pair-collision error `seeds(w) - (nmax - sum_{q>w} N_q) = sum_{i=1}^k C(b_i - 1, 2)` is owned by the cluster-graph cyclomatic number (Bollobas 1998, Ch. I Sec. 2 Thm. 4, p. 10; Feller 1968 Ch. IV) and internally by Return #5462. Verified across all 152 `(x, w)` cells (`11#..71#`), with positive excess on 20/152 cells. 2. **Exact Residual-Hole Variety Proof at 43# vs. 59# (`C06-C09`):** At `43#` (`G2=618, nmax=8`), each `T_19` seed (`8521991` and `1177079`) has 3 uncovered interior holes `{156, 240, 450}` (resp. `{168, 378, 462}`) after `{23, 29, 31, 37}`. Because `240 - 156 = 84 == +2 (mod 41) == -2 (mod 43)`, either `41` covers `{156, 240}` at `r_41 = 6` while `43` covers `{450}` in `2` ways (`{21, 23}`), or `43` covers `{156, 240}` at `r_43 = 16` while `41` covers `{450}` in `2` ways (`{1, 40}`), forming the orthogonal cross `({6}x{21,23}) U ({1,40}x{16})` with `3` mod-41 residues per seed (`seeds(41) = 6`, `N_41 = 10`, `N_43 = 2`). At `59#` (`G2=966, nmax=8`), floating `{43, 59}` face holes `{198, 240, 906}` with `906 - 198 = 12 * 59`, forcing `59` to cover `{198, 906}` (`{36, 38}`) and `43` to cover `{240}` (`{16, 18}`), a `2x2` Cartesian box (`seeds(43..53) = 4`, `N_43 = 8`, `N_59 = 4`). 3. **Complete 11#..71# Ladder Including New 67# and 71# Tie Sets (`C10-C12`):** All 104 positions across `11#..71#` verified by direct `gcd`; at `67#` (`G2=1284`), `nmax = 2`; at `71#` (`G2=1398`), `nmax = 4` with floating prime `q* = 37` covering `{570, 792}` (`792 - 570 = 6 * 37`, `N_37 = 2`).