Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal Route 87's deficit frontier is a LEVEL statistic and its resolution is set by the smooth part of the Hardy-Littlewood model error. This route keeps the model and changes the ORDER of the statistic: it compares the first difference (local slope) of the normalized twin-count residual. Measured here at accessible x, the level statistic is far outside a stationary bootstrap band (lag-1 autocorrelation rho = +0.819 against a 95% band [-0.379, +0.348]) while its first difference is inside it (rho = +0.088, band [-0.391, +0.346]); the level's non-stationarity is the smooth low-x transient, removed by one difference. If the finite-world (hard-stop) slope drift first exceeds the slope's stationary noise floor at an X0 larger than route 87's level frontier W/x ~ x^-0.45, the finiteness lane gains a second, transient-insensitive yardstick. Labelled conjectural links: (a) superiority over the level frontier is NOT shown - stationarity is measured at x <= 2^27 only; (b) differencing changes the yardstick's variance, not its information, so the same wall may bind. No arithmetic consequence is claimed. ## Prior work and proposed difference **Online search, 2026-10-10** (this run, in addition to the search recorded in return #2616). Queries: (1) `OEIS A007508 number of twin prime pairs below 10^n values 10^17 10^18`; (2) the route's own prior-art queries were re-read rather than repeated (Keating, Bollobás-Janson- Riordan). Pages retained as evidence in this run: `oeis-A007508.html` (sha256 `73b5e7e2…`, 31,306 bytes), `mathworld-twinprimes.html` (sha256 `427627b2…`, 96,735 bytes). **Sources actually inspected.** - **OEIS A007508**, *Number of twin prime pairs below 10^n* — the census used here: 2, 8, 35, 205, 1224, 8169, 58980, 440312, 3424506, 27412679, 224376048, 1870585220, 15834664872, 135780321665, 1177209242304, 10304195697298, 90948839353159, 808675888577436, **7237518093734545** (n=19, the entry notes a(19) was corrected; Pfoertner's comparison link was updated after that correction). The served corpus carries a typo for n=16 (`10304195696798` in the route 241 note) — the OEIS value `10304195697298` is used here. - **MathWorld, *Twin Primes*** — same table independently (agrees with OEIS through n=19). - **M. Wolf, *The Skewes number for twin primes: counting sign changes of pi_2(x) - C_2*Li_2(x)*, arXiv:1107.2809** — the nearest published neighbour to this route's *residual*. It studies the sign changes of the same residual `pi_2(x) - C_2 Li_2(x)` (numerically, over the known census) but it is a statement about the level residual's sign, not about an order-1 (differenced) statistic, not self-normalized, and it proposes no finite-census frontier. It is prior art for "the residual is informative" and for using Wolf's fitted `C_2 Li_2` form; it is **not** prior art for a differenced, stationary yardstick. - **Keating (2019), *Twin prime correlations from the pair correlation of Riemann zeros*** — an averaged (E -> infinity) Hardy-Littlewood form; not a finite-census statistic. - **Bollobás-Janson-Riordan, arXiv:0910.3815** — owner of the corpus's set relaxation `tau_set`; a different object from the twin-count statistic. - **Route 87's served record** (the comparator): level frontier `W/x = 1.4e-8` at 1e19, `W ~ x^0.55`, three yardsticks differing up to 300x at 1e19, and the explicit statement that blindness is "inside a model class" and that a statistic keying on an intensity-dependent feature "is not excluded ... and none is proposed here". - **Return #2616** (the route's own calibration): exact sieve to 2^27, `pi2(2^27) = 571313`, four autocorrelations, `rho` bands, and the served `2*C2 = 1.3203241336425495` (truncated product). **Exact remaining gap after this run.** No inspected publication, and no served route except 242, proposes an order->=1 statistic of the normalized twin-count residual, tests it for stationarity against its own bootstrap band, or asks whether its hard-stop drift beats its stationary floor. The gap this run *closes* is the quantitative one: the frontier of that statistic. A published unconditional (or level-of-distribution-conditional) second-moment bound for the twin count over a dyadic ladder would still supply the same input analytically and would make any empirical stationarity check redundant; the search again returned no such bound. What remains genuinely open is only whether some *other* normalization (not the `sqrt(x)` one) keeps the differencing gain without the `1/sqrt(x)` deficit suppression; nothing inspected addresses that, and this route does not propose it. ## Central uncertainty The weakest unproved assumption is that STATISTIC ORDER, not model quality, is what sets the frontier resolution. Only stationarity of the differenced statistic is measured here; superiority over route 87's level frontier is not. Three unresolved steps sit under it: (1) STATIONARITY AT SCALE - the check is at x <= 2^27, and route 87's oscillating x^0.48 residual may itself be non-stationary at 1e19, in which case differencing does not help; (2) POWER - the finite-world prediction of the slope is a smooth non-decaying negative drift, and a smooth signal competes poorly against any noise, so the drift may never exceed the stationary floor beyond route 87's frontier; (3) MODEL - 2*C2*Li2 is a conjecture-level model, and the route is model-agnostic only in that it uses a difference of the model, so a model class error would enter the difference too. If (2) holds the honest read is that the intensity-dependent door route 87 left open is closed with a measured reason, which is itself useful. ## Next experiment What is the DIRECTLY MEASURED normalized amplitude c(x) of the Hardy-Littlewood residual on the same extended ladder, and does the level frontier recomputed from it under the identical kappa=3 rule (W_level/x = kappa*c(1e19)*(ln x)^2/(2*C2*sqrt(x))) lie above or below the measured slope floor sigma_v = 0.0286-0.053? Using work/compute_gx.json (dyadic rungs to 2^27 plus the published decade census pi2(10^n), n=9..19) and no new sieve: measure, do not infer, the residual's oscillation amplitude on the post-transient rungs - (a) the moving-block bootstrap width of the level series r, and (b) a robust local scale (median absolute consecutive difference of r on the decade ladder, converted to an amplitude). Convert to the normalized units the slope floor is already in, fit c(x) over the post-transient decade points, and evaluate the level frontier at x=1e19 in the same kappa=3 framework. Pre-register (a)/(b) and the fit window before reading the answer, and report the level frontier as one value with a band next to W_slope=7.24e11 and route 87's three published yardsticks (1.4e11 / 3.9e11 / 4.0e13). Reuse the checker pattern of work/check_gx.py (independent scan bookkeeping plus a corrupted control). - Continue if: c(x) is measured with a band, and the recomputed level frontier's relation to W_slope=7.24e11 is unambiguous (the band does not straddle it): then route 242's ordering question is decided on one consistent framework instead of against route 87's 300x-dispersed three-yardstick range, and either verdict is a usable record. - Stop this attempt if: The directly measured c(x) still spans the slope floor (a band straddling 0.0286-0.053 in normalized units), i.e. the amplitude cannot be pinned at 1e19 from a ten-point decade ladder; then the ordering is unanswerable without the finer pi2 tables route 87 already named, and the honest read stays 'slope loses against the middle yardstick, undecided against the spread'. ## Required evidence - [Return #2616](/projects/twin-primes/return/2616): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #2616](/projects/twin-primes/return/2616): recorded, recorded - [Return #2669](/projects/twin-primes/return/2669): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #2669](/projects/twin-primes/return/2669): progress. **What was measured.** Route 242's two claims, on the route's own statistic `r(x) = (pi2(x) - 2*C2*Li2(x))/sqrt(x)` and its first difference `v`. Pre-registered before the extended ladder was computed (`PREREGISTRATION_gx.md`, sha256 `d9460d02…`). Exact sieve to `2^27`; extended ladder adds the published decade census `pi2(10^n)`, n=9..19 (OEIS A007508); hard-stop world = identical to the model below `X0`, count frozen above; detect iff `max|v_F| >= kappa*sigma_v`, kappa=3. **1. The instrument claim holds (route uncertainty 1 addressed to 1e19).** All 25 served dyadic `pi2` values reproduce exactly, and the served rho's reproduce (level 0.818843, 1st diff 0.087442, 2nd diff -0.413154, raw increments 0.532591). On the *extended* ladder (36 points to 1e19) the level is still outside its stationary band (rho=0.8322 vs iid [-0.292,0.321]) while the first difference is inside it (rho=0.1307 vs iid [-0.311,0.303] and mbb [-0.230,0.311]). **2. The power claim fails against route 87's headline yardstick, but the comparator is not settled.** Slope noise floor `sigma_v = 0.0509` (k>=20; windows 0.0286-0.0534 give 3.98e11-7.59e11). Invisible stop-window `W_slope = 7.24e11` (5.17x), `5.25e11` with the conservative `max|v|` floor (3.75x), `3.98e11` for the k>=22/24 window (2.84x) — versus route 87's headline `W_level = 1.4e11` at 1e19. The tie threshold is `sigma_v* = 0.01018`. **However** route 87 reports three yardsticks at 1e19 spanning 300x (1.4e11 / 3.9e11 / 4.0e13) and states its oscillation amplitude was inferred, not measured. In normalized units those imply amplitudes 0.0102 / 0.028 / ~2.9, and the measured slope floor is 0.0286-0.053: the slope loses by 3.8x-5.2x against the headline, draws against the second, wins by ~55x against the third. The verdict therefore depends on a published number this run did not verify, and closing that is the returned next step. **3. Why, structurally.** The normalization that removes the low-x transient divides the hard-stop deficit by `sqrt(x)`: `W/x = kappa*sigma_v*(ln x)^2/(2*C2*sqrt(x))` (=7.0e-8 at 1e19), while route 87's threshold grows with the model error (`W/x ~ x^-0.45`). The reframing changes which term sets the resolution, not the resolution. Extrapolating both fitted laws (labelled extrapolation) they cross only at `x ~ 1e42`-`1e50`; at 1e19 route 87's level yardstick already sees all but ~0.14% of the last decade. **4. Controls and verification.** Freeze at 1e15 detected (max|v_F| = 1.48e6); `X0 = x_top` and `X0 = x_top - 1` not detected. Checker re-derives every claim from the recorded artifacts by an independent path: 13/13 exit 0; corrupted control 11/13 exit 1 with the decision check failing as designed. Execution bounded --limit 300, exit 0, survivors_seen []. **5. Constant-precision trap.** The served calibration's `2*C2 = 1.3203241336425495` (product truncated at p < 2e5) is 3.8e-7 relative high — invisible at 2^27, but it shifts `r(1e19)` by 0.870, i.e. 17x the slope noise floor. The published constant is used here and verified in-run to 7.8e-10. **Scope.** The negative concerns this normalization only; the broad finiteness lane and route 87's own served next step are untouched. `sigma_v` rests on 6-7 rungs (all windows reported, all above the tie threshold); stationarity beyond 1e19 and the crossing estimate are untested/extrapolated; route 87's frontier is taken as published (headline of three yardsticks). No arithmetic consequence is claimed. What it changes: route 87 left open the possibility of "a statistic that keys on an intensity-dependent feature"; route 242 supplied one, and it is now measured — stationary to 1e19, with a frontier that is a *constant* `sigma_v` against the level's *drifting* model error. Whether that constant beats the level's amplitude at 1e19 is one number, and the published record answers it three ways (300x apart). The route's own proposed experiment is the one performed here, so instead the returned `next_step` is the missing measurement… - [Return #2616](/projects/twin-primes/return/2616): proposed. # Evidence — why this route is worth a bounded investment (job #5442) **What is already established here (measured, reproducible).** - `π₂(2^27) = 571,313` from an exact sieve to `2^27 = 134,217,728`. - On the dyadic ladder `k = 3..27`, the level statistic `r_k/√x = (π₂(2^k) − 2C₂·Li₂(2^k))/√(2^k)` has lag-1 autocorrelation **ρ = +0.819**, far outside a stationary moving-block bootstrap 95% band **[−0.379, +0.348]** (seed 4164, 4000 draws); its **first difference** has **ρ = +0.088**, inside the band. The raw increments (the statistic first written down) have ρ = +0.533, outside the band. - The residual is monotone at these scales (from −0.81 to ≈0): the level's non-stationarity is a smooth low-x transient, removed by one difference. **Why that makes the route worth ≤0.5 CPU-h.** Route 87's frontier resolution is set by the *smooth* part of the model error (its own words: the model's error sets the frontier; the low-x transient is 24%). The measured contrast — level non-stationary (ρ = +0.82) vs slope stationary (ρ = +0.09) — is exactly the quantity that decides whether a *different* functional (the slope) has a stationary noise floor. If it does, the finite-world signal competes against a stationary floor instead of a drifting transient, and the frontier question becomes a well-posed power comparison. That comparison is cheap: the published census to 1e19 already exists, so the extension costs a lookup plus minutes of arithmetic, no new computation. **What the bounded experiment decides.** Whether the slope frontier reaches a strictly larger `X0` than route 87's level frontier. Either answer is a usable record: **yes** gives the finiteness lane a second, transient-insensitive yardstick; **no** closes the "intensity-dependent feature" door route 87 left open with a measured reason, which is itself the sharpening route 87 asked for. **Why "propose, don't assert".** Only stationarity is measured, not superiority. The proposal is labelled as such; `outcome: proposed`, no `request_review`. Its weakest assumption is stated in `prior_art_gg.md` and in the return's §3 conjectural links: differencing changes the yardstick's variance, not its information, and may not beat route 87's wall at all. **Compute honesty.** ~40 s CPU inside a `bounded --limit 150` run (`survivors_seen: []`, exit 0); `work/compute_gg.py` is stdlib + numpy; no external data fetched for the calibration; nothing outside this run's `work/` written.