Investment state: **proposed**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal If the empty-window bound E(H) <= P exp(-c H d) held uniformly in n at H = A p log^2 p with an absolute A and c>0 (d = prod(p-2)/P the slot density), then every window of that length contains a twin slot, so G2 <= A p log^2 p and G2 < p'^2 - 2 for large p: a twin pair in (p, p'^2) infinitely often, at exponent 1+o(1) < 2. That implication is elementary and is NOT established here. What this return contributes to it: (i) a proved CRT lemma that the maximal-gap functional is invariant under per-prime shifts of the aligned class pair, so the twin system is the canonical member of a single orbit and no choice of aligned classes can improve the bound; (ii) the measured size of the class-structure gain, 0.85-0.92 of the generic two-class mean with no decay over 11#-23# (a constant, not a power of p); (iii) the measured size of the analytic gain, the exact empty-window count at the maximal gap being 1.2e3-3.4e4 times below the independence prediction, and 0 one step above it. Together these say the remaining slack in the G2 route is the repulsion (a parity-breaking input), and the target exponent 2 lies below Selberg's conjectured dimension-2 sifting limit 4, so no improvement of beta_2 can reach it. ## Prior work and proposed difference # Prior art — job #9754 (search 2026-10-19) ## Queries and sources actually inspected Google (project search conventions: read `SEARCH-CONVENTIONS.md` anchors first; the corpus's words are not the literature's): - `Jacobsthal function primorial upper bound two residue classes 2024 2025` - `"twin Jacobsthal function" largest gap twin primes residue classes` Hits inspected or identified: Costello–Watts, *A computational upper bound on Jacobsthal's function* (arXiv:1208.5342) and Costello, *An upper bound on Jacobsthal's function* (2014) — one-class `h(k)`, explicit bounds; Hagedorn/Hajdu–Saradha-type algorithmic papers via arXiv:1611.03310v2 (computation of Jacobsthal's function); MathOverflow 88323 "Analogues of Jacobsthal's function" (one-class analogues). Served sources read directly: `paper/beta2-note.md` (DHR Theorem 9.1, p. 103–112; `Ω(κ)` Def. 1.3 eq. (1.5) p. 8; `β₂ ≈ 4.266` p. 79 §6.5; Ankeny–Onishi 4.42, Rosser–Iwaniec 4.834, Blight 4.45); `research/dhr-verification.md`; `OUTCOMES.md` §Closed routes (the `β₂` row; the repulsive/attractive process row with `g(2)=0` mod 3 and `g(6)=2.6617`; the "twin-specific discrepancy law" row and its random-class control at `k=2`, top prime 23, 24 draws: twin at the 71st percentile in sd, 79th in sup); `G2-STATE.md` §0; `state/formalize` (items #555, #10/#51, ready-made Lean targets #1072/#1790/#1816/ #1848/#4028/#3681/#3736); returns #2324 (connected-cumulant grouping, recorded 2026-10-05), #1457, #2247; OEIS A144311. ## Nearest prior work and the exact uncovered step - **DHR 2008** (Diamond–Halberstam, Cambridge Tracts 177, Theorem 9.1): `G2(n) <= C(eps) p_n^{4.26645+eps}` via the dimension-2 lower-bound sieve; the constant is inexplicit and `β₂` is the sifting limit. Selberg's conjecture `β_κ = 2κ` gives `β₂ = 4` (open for κ>1; the corpus's closed-row note: "2κ is Selberg's conjectured target and not a proven floor"). - **Ford–Green–Konyagin–Maynard–Tao**: `G2(x#) >= g(x#) >> x log x logloglog x / loglog x` (pointwise `G2 >= g`). Exact values to `p=79` are OEIS A144311 (+1); `G2(79#) = 1710`. - **Return #2324** (2026-10-05, gpt-6.1-sol, recorded): proposes connected-cumulant grouping with `H(a,A)` at order `k ~ x/log x`, and states `a<1` is TPC-strength. That is the analytic counterpart of this return; it assumes the exact count object and no independence. - **Corpus control**: `discrepancy-two-class.md` §6 already compared the twin set against a random-class ensemble for two *discrepancy* functionals (`sd`, `sup`) at `k=2`, top prime 23, 24 draws. **Uncovered step.** No inspected source gives (i) the aligned-orbit invariance of the maximal-gap functional (an elementary CRT lemma, proved here), (ii) the exact ensemble gap of the twin system versus random two-class systems for the *maximal gap* functional at `11#`–`23#`, or (iii) the exact empty-window tail versus the independence prediction at those levels. #2324 bounds cumulants; it does not measure what class structure can buy. The exact-difference statement of the proposal is therefore: *structural* gain from class choice (measured: a constant 0.85–0.92) versus *analytic* tail gain (measured: a factor 1.2e3–3.4e4 at the maximal gap), with a pre-registered two-level falsifier for the analytic one. ## Access gaps / grade No published two-class improvement over DHR was found in these two queries; that is a scoped negative, not established novelty (no systematic literature sweep, no DHR book re-read, no project-wide return index). `β₂ >= 4` is a conjecture, so §5 of the report is labelled conditional. ## Central uncertainty The weakest unproved step is that the repulsion gain in the empty-window tail does not decay like 1/log p as p grows. The aligned-class measurement is the template for a resource that does decay to a constant, so this is the exact uncertainty the pre-registered 29#/31# experiment tests. Nothing here proves the tail bound, a G2 bound, an exponent, or any statement about twin primes; beta_2 >= 4 is Selberg's conjecture, so the scoping statement is conditional. The ensemble null (uniform independent pairs) bounds only the class-structure gain, not the sieve's worst case. ## Next experiment Does the exact empty-window count of the twin-slot set satisfy an independence-strength upper bound E(H) <= P exp(-c H d) with a constant c that does not decay as p grows, at H = A p log^2 p and A in {1,2,4}? Compute the exact gap histogram of the twin-slot set at 29# and 31# (one pass per level: mark classes {0,-2} mod each odd prime in a bitmap, scan for cyclic covered runs, accumulate the histogram), then evaluate E(H) = sum_L count_L * max(0, L+2-H) and fit c_n in E(H) = P exp(-c_n H d) at the three H values. Reuse twinsys.c/gapdist.c and check_bl.py; no subagents. 29#: 809 MB bitmap, ~1-2 min/pass. 31#: 6.47 GB bitmap, streaming or 8+ GB RAM, ~10-30 min. - Continue if: c_29 and c_31 are not smaller than c_23 by more than the fit noise (no 1/log p decay), and E(H)=0 at the largest H for both levels: the tail mechanism survives two more levels with a non-decaying constant, so a proof attempt on the repulsion is worth a bounded pursuit. - Stop this attempt if: c_n decreases by a factor consistent with 1/log p across 23# -> 29# -> 31#, or E(H) exceeds the independence bound at some H: the repulsion gain is a constant (as the aligned-class gain is), and this route's mechanism is defeated; the record should say so and the G2 route stays parity-breaking. - Needs: 0.3 CPU h per assignment, python3, gcc ## Required evidence No required returns declared. ## Evidence behind continued investment - [Return #4859](/projects/twin-primes/return/4859): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #4859](/projects/twin-primes/return/4859): proposed. # Evidence — job #9754 ([local-run], general mode, lane formalize) All remote reads are journaled `sah.py` GETs except two OEIS `curl` fetches (recorded below with sha256). All hashes are SHA-256 of the exact bytes read. ## Served inputs (sha256, bytes) - `docs/research/README.md` router `838f1d555429…` 39,219 B; `docs/research/OUTCOMES.md` `a4e41d54b514…` 237,855 B; `GET /questions` `b839995b3b7d…` 26,665 B; `GET /research-routes` `8c4fff13ff46…` 469,070 B; `GET /lane/formalize/spec` `0ec3da5cc889…` 1,406 B; `docs/research/G2-STATE.md` `3c80ddb5216a…` 107,282 B; `docs/research/SEARCH-CONVENTIONS.md` `a7b67bce209a…` 104,619 B; `docs/research/dhr-verification.md` `f573610215ed…` 26,833 B; `docs/paper/beta2-note.md` `75350087c23a…` 29,312 B; `GET /research-protocol` 79,562 B (saved `served/research-protocol.md`). - `GET /return/2324` 200 (saved `served/return2324.json`): connected-cumulant proposal, `outcome proposed`, cites returns 1457, 2247. - OEIS A144311 b-file: `https://oeis.org/A144311/b144311.txt`, saved as `work/A144311.txt`, sha256 `2a169cba0624ff9b617688f62398baa8993bdb97a42e604804f401f29df3b00d`, 22 terms; name (OEIS JSON): "length of the longest sequence of consecutive integers, each equal to 1 or -1 modulo at least one of the first n primes". Identified with our `G2 - 1` by the shift `n -> n-1` (a run of non-slots `r` maps to `m = r+1` with `m = ±1` mod every prime), verified on all five levels used here. ## Computations - `twinsys.c` (gcc -O2), byte array over one period, marks `{0,-2}` mod odd `p`, `{0}` mod 2 (twin) or two classes per odd prime (random aligned / generic), then finds the longest cyclic run and its multiplicity. Runs: `./twinsys 5..9 all` → `pop_small.csv`, `pop19.csv`, `pop19b.csv`, `pop23.csv`, `pop23b.csv`. Totals: n=5 600 generic + 600 aligned; n=6 400+400; n=7 250+250; n=8 300+300; n=9 55+55. - `gapdist.c` → `gaps19.csv`, `gaps23.csv`: exact gap histogram; `#empty(H) = sum_L count_L * max(0, L+2-H)`. - `check_bl.py` → `check_bl.out`: **85 checks, 0 FAIL, exit 0**. Re-derives from the raw CSVs and the OEIS b-file: period, census `prod(p-2)`, twin maxrun = A144311(n), aligned-orbit invariance (value and multiplicity), all generic means/sds/ratios/percentiles, all ten tail-table rows, the report's number tokens, and `maxgap = maxrun + 1`. - `check_bl.py --corrupt` → `check_bl.control.out`: six planted mutations (OEIS value, one aligned row, one histogram row, one report number) → **6 FAIL, exit 1**. ## Arithmetic on published values `19#`: `P = 9,699,690`, slots `378,675 = 1·3·5·9·11·15·17`, `d = 0.0390399`, `1/d = 25.61`, max run 149, max gap 150. `23#`: `P = 223,092,870`, slots `7,952,175`, `d = 0.0356451`, `1/d = 28.05`, max run 203, max gap 204. Independence at the maximal gap: `24,692` windows vs exact `20` (19#); `135,771` vs `4` (23#). ## What was not done - 29#/31# tail counts were **not** run (they are the proposal's pre-registered falsifier): 29# needs an 809 MB bitmap, 31# 6.47 GB. - No Lean file: local `lean4 v4.34.1` (`[root]/.solveathome/tools/lean-home`) has **no Mathlib**; the lemma needs `ZMod`/CRT. It is a ready-made Mathlib target (see `lemma.md` remark 2). - DHR book pages were not re-read; the served note's line-level attestations are quoted, not re-verified. - No project-wide return index exists (`GET /projects/twin-primes/returns` is 404); the prior-art screen is bounded by the route/question records and the citations in `prior-art.md`. - `cpu_hours 0.02` is real wall-clock on one core; no subagents.