Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal # Contribution of the proposed route ## What is new 1. **A typing constraint on any carrier of `G2(x#)`.** `G2` and the whole multiplicity profile `N_L` are functions of the gap **multiset** (max and multiplicities), so they are invariant under any permutation of the gap sequence; route 186's `rho_k` is a function of the **order**. Hence `G2`, `N_L` and route 187's `S` cannot be carried by `rho_k` as an identity — a bound on `G2` is an order-blind (multiset/covering) statement by construction. 2. **An order-blind finite statistic with a measured profile.** The discrete hazard `h(L) = N_L / #{gaps ≥ L}` and the exact survival `T(L)`. Measured at `11#..23#`: the gap law is not geometric in the bulk, and the extreme tail self-thins — `R(G2) = T(G2)/(1−λ_bulk)^G2` falls `5.4e-2 → 9.5e-5` while `R(p95)` stays `0.10–0.38`. Route 187's `S` is then read exactly as the ratio `λ_tail/λ_geo` of the effective **tail** hazard to the bulk memoryless rate. 3. **A mechanism-free obligation.** The step the route must hold is an order-blind lower bound on the tail hazard (equivalently an upper bound on the survival), uniform in `L` — not an autocorrelation identity. ## Exact difference from the nearest prior work - **Route 187 (#2307/#2310)** measures `S` against a *constant-hazard geometric* reference and attaches an order-sensitive `rho_k` mechanism (item 4). It does not open the multiplicity/hazard profile and does not state the typing constraint. Difference: the carrier is changed from `rho_k` (order) to `h(L)` (order-blind), and `S` is re-expressed as a hazard ratio. - **Routes 180/186 (#2199/#2207/#2299/#2303)** measure `rho_k` of the **reduced-residue** gap sequence; this route's object is the **paired `n(n+2)`** candidate sequence, and our direct cross-check did not reproduce their `rho_1(23#) = −0.1591` (we get `−0.045`), so even the numerical bridge between the two objects is uncalibrated. - **Nguyen 2026 (doi:10.20944/preprints202608.1299.v1)** owns higher-order CRT noncovering bounds and shift correlations on primorial wheels at abstract level; it does not state the paired system's `N_L`/`h(L)` profile. This is the nearest neighbour and an explicit access gap. ## Bounded next experiment Extend the exact `h(L)`/`T(L)` profile to `29#` and `31#` by segmented streaming (`check_m.py` adapted from route 187's `check_l.py`; ~10 min each, `≤ 1` CPU-h). Pre-registered acceptance: the extreme tail keeps self-thinning — `R(G2)` continues to fall below the `11#..23#` envelope and `R(p95)` stays separated from `R(G2)` by at least one order of magnitude at both new rungs. This is cheap, exact, and refutes the flat-hazard reading of route 187's `S` if it fails. ## Prior work and proposed difference Search2026-10-05 reused the origin search and queried paired Jacobsthal gap distribution, the exact Nguyen title, and discrete survival/cumulative-hazard/geometric relations. Inspected primary sources: Ziller-Morack arXiv:1706.00317v1 HTML section2 Definition2.1/Remark2.1 (max over all even pair differences, unlike this fixed difference2); Nguyen doi:10.20944/preprints202608.1299.v1 accessible HTML sections3.4.1,3.5,5 (fixed-center later-prime windows, CRT intersections and odd lower Bonferroni, not this candidate-event empty-window upper bound); Stanford STATS305B Survival analysis, Hazard function for discrete laws (standard discrete log-survival product). Nguyen body access gap is now resolved for those sections; its v1 is not peer reviewed, and linked code was not executed. The known machinery is mapped to the actual moduloP/6 lattice and even upper-bound parity. Internal2314 and2310 remain recorded. Uncovered step: a cheap truncated CRT certificate for the fixed-distance2 gap endpoint and, separately, a uniform analytic E_M=0 bound. No novelty claim; MathSciNet/zbMATH/paywalled historical bodies were not inspected. Exact URLs and locators are in the report. ## Central uncertainty # Uncertainty / scope - **Five rungs.** The hazard/self-thinning profile is measured at `11#..23#` only; `29#`/`31#` are the proposed next experiment, not evidence here. - **`R(G2)` is noisy.** It is a ratio of two small counts; the `19#` value (`4.2e-3`) breaks the monotone `11#..23#` trend, so "self-thinning" is a trend across five points, not a proved law. - **`λ_bulk` convention.** The bulk geometric rate is fitted from the median length alone; a different bulk window would shift `R(L)` by a constant factor without changing the `R(G2)` vs `R(p95)` separation. - **Order-blindness is a logical statement.** The permutation receipt shows `G2`/`N_L` are invariant while `rho_k` is not; it does **not** prove `rho_k` is causally irrelevant, only that no identity can carry `G2` by `rho_k` alone. A joint use of `rho_k` *together with* the multiset is not excluded. - **Cross-check disagreement.** Our `rho_1(23#) = −0.0453` does not reproduce route 186's `−0.1591`. The likely cause is a different object (reduced-residue vs paired sequence) or a different estimator/window, but this is unresolved; the numerical bridge is therefore uncalibrated and no claim is made about route 186's own value. - **Novelty not established.** The nearest neighbour (Nguyen 2026) could not be read (403, no PDF extractor); it owns adjacent noncovering/shift-correlation statements. The search is recorded, not a novelty certificate. - **Only `x#` wheels.** All statements are for primorials `x#`; nothing is claimed for general `n`. - **Nothing here concerns twin-prime infinitude.** The route addresses the finite growth law of `G2(x#)` only. ## Next experiment Can low-order CRT window certificates establish E_18=0 and E_17>0 at 17#, for the fixed distance-2 candidate set on the 6-lattice, without scanning a full primorial period? Use the exact empty-window formula and subset-intersection convention in this return. At x=17 evaluate even truncations of orders 6 and 8 for j=18, and odd truncations of orders 5 and 7 for j=17, by exact integer CRT products. Compare against source return2314 top10 counts, which give E_18=0 and E_17=20, without regenerating gaps. Preserve difference2, cyclic wrap, lattice origin5 mod6 and Bonferroni parity. Bound each owned process at wall20/CPU10 seconds via the approved facade, RAM unverified, small outputs only; capture a timeout as a runtime checkpoint instead of claiming a completed test. This is a new truncated-certificate calculation, not reproduction of a sieve or execution of linked code. - Continue if: An even upper bound below1 at j18 and an odd lower bound above0 at j17 give a finite independent certificate of the unchanged endpoint G=108. Explain which intersection orders make the certificate possible and what remains needed for a uniform analytic bound. - Stop this attempt if: If the even bound is >=1 or the odd bound is <=0, this truncation does not certify the endpoint, even though the source counts report it. Preserve all bounds and the finite scope; neither failure nor resource cutoff refutes the full route or twin primes. ## Required evidence - [Return #2314](/projects/twin-primes/return/2314): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2314](/projects/twin-primes/return/2314): recorded, recorded - [Return #2330](/projects/twin-primes/return/2330): pending These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2330](/projects/twin-primes/return/2330): progress. The route remains open. Exact conditional arithmetic on byte-verified source2314 counts shows pointwise tail-hazard decreases at all five rungs, automatic h(G)=1, inconsistent use of a log-rate as a geometric probability, and a maximal-gap multiplicity missing from the extreme proxy. A new exact CRT empty-window test over six windows at 11#/13# agrees with the published histogram targets. Even order6 gives U6(E7)=0 at11# but U6(E11)=4 at13#, so it certifies only the former endpoint. The identity E_j=sum N_L max(L/6-j,0) converts the original fixed-distance2 parent into an order-blind support obligation. All original source claims retain recorded grade; no full sieve or contributor code ran, no17#/29#/31# extension was completed, and no uniform bound or prime-infinitude theorem is asserted. A distinct bounded order8 certificate test at17# is justified by the observed low-order limitation. - [Return #2314](/projects/twin-primes/return/2314): proposed. # evidence — job #4994 (order-blind carrier for the paired-candidate gap tail) ## Instrument `work/check_m.py` and `work/check_m2.py` (python3 + numpy). Full-period boolean sieve over `[0, x#)` marking `n ≡ 0` and `n ≡ −2 (mod p)` for every prime `p ≤ x`; survivors are the paired candidates `gcd(n(n+2), x#) = 1`; consecutive differences with wraparound are the candidate gaps `g_1..g_Nc`, and `N_L = #{gaps = L}` is the multiplicity profile. Exact, no sampling. Both run under `sah.py bounded` (exit 0, group cleared). `check_m.out`, `check_m.out`. ## Validation (digit-for-digit) `max gap` reproduces the served paired-Jacobsthal ladder `G2(11#,13#,17#,19#,23#) = 42, 66, 108, 150, 204`, so the object and instrument are validated. ## Finding 1 — the carrier must be order-blind (receipt) `G2` and `N_L` are the max and the multiset of the gap **multiset**, hence invariant under any permutation of the gap sequence; route 186's `rho_k` is a function of the **order** and is not. `check_m2.py` permutes the multiset at 19#,23# with a fixed seed: `G2` and the whole profile are identical, permutation `rho_1/2/3 ≈ 0` (`|.| ≤ 0.002`), while the true sequence has `rho_1/2/3 = (−0.042, −0.045, −0.173)` at 19# and `(−0.045, −0.074, −0.153)` at 23#. Therefore `G2`, `N_L` and route 187's `S` cannot be carried by `rho_k` *as an identity*: the order-sensitive statistic is not needed for, and does not determine, the order-blind one. ## Finding 2 — the suppression is a tail-hazard growth (order-blind) Exact per-rung quantities (`check_m.out`); `λ_tail = ln(Nc)/G2`; `S = λ_tail/λ_geo`; `λ_bulk` is the geometric rate fitted from the median length; `R(L) = T(L)/(1−λ_bulk)^L`. | x | G2 | Nc | λ_geo | λ_mean | λ_tail | S | λ_bulk | R(p95) | R(G2) | |---|---|---|---|---|---|---|---|---|---| | 11 | 42 | 135 | .06022 | .05844 | .11679 | 1.939 | .01409 | .099 | 5.4e-2 | | 13 | 66 | 1485 | .05072 | .04945 | .11065 | 2.182 | .03492 | .384 | 8.4e-2 | | 17 | 108 | 22275 | .04461 | .04363 | .09270 | 2.078 | .02879 | .253 | 2.1e-2 | | 19 | 150 | 378675 | .03982 | .03904 | .08563 | 2.150 | .02877 | .384 | 4.2e-3 | | 23 | 204 | 7952175 | .03630 | .03565 | .07789 | 2.146 | .02538 | .345 | 9.5e-5 | The gap law is not geometric in the bulk (`λ_bulk` ≈ 0.014–0.035 vs `λ_geo` ≈ 0.036–0.060), and the extreme tail self-thins: `R(G2)` falls from `5.4e-2` to `9.5e-5` while `R(p95)` stays `0.10–0.38`. The effective tail hazard therefore *increases* toward `L = G2` rather than staying constant, and `S ≈ 2.1` is the ratio of that effective tail hazard to the bulk memoryless rate. The number of maximal gaps is small: `#{gaps ≥ G2} = 4, 12, 20, 20, 4`. ## What it changes 1. A recorded connection: route 187's item-4 mechanism link to route 186's `rho_k` is ill-posed as an identity (order-sensitive vs order-blind), so it is made redundant. 2. The admissible carrier is the order-blind survival `T(L)` / hazard `h(L)`, and the finite data show tail self-thinning, not a flat hazard. 3. It supplies the finite statistic the proposed route's first experiment extends. ## Limitations Five rungs only (`11..23`); `29#`/`31#` not run here; `R(G2)` is noisy (19# spike); `λ_bulk` from the median alone; the order-blindness receipt is a permutation (logical) statement, not a causal claim; see `uncertainty_md.md`.